6.02 Practice Problems: Noise & Bit Errors

Transcription

1 1 of Practice Problems: Noise & Bit Errors Problem 1. Suppose the bit detection sample at the receiver is V + noise volts when the sample corresponds to a transmitted '1', and noise volts when the sample corresponds to a transmitted '0', where noise is a zero-mean Normal(Gaussian) random variable with standard deviation σ NOISE. A. If the transmitter is equally likely to send '0''s or '1''s, and V/2 volts is used as the threshold for deciding whether the received bit is a '0' or a '1', give an expression for the bit-error rate (BER) in terms of the erfc function and σ NOISE. B. Suppose the transmitter is equally likely to send zeros or ones and uses zero volt samples to represent a '0' and one volt samples to represent a '1'. If the receiver uses 0.5 volts as the threshold for deciding bit value, for what value of σ NOISE is the probability of a bit error approximately equal to 1/5? C. Will your answer for σ NOISE in part (B) change if the threshold used by the receiver is shifted to 0.6 volts? Do not try to determine σ NOISE, but justify your answer. D. Will your answer for σ NOISE in part (B) change if the transmitter is twice as likely to send ones as zeros, but the receiver still uses a threshold of 0.5 volts? Do not try to determine σ NOISE, but justify your answer. Problem 2. Messages are transmitted along a noisy channel using the following protocol: a "0" bit is transmitted as -0.5 Volt and a "1" bit as 0.5 Volt. The PDF of the total noise added by the channel, H, is shown below. It is not a Gaussian.

2 2 of 4 a. Compute H(0), the maximum value of H. b. It is known that a "0" bits 3 times as likely to be transmitted as a "1" bit. The PDF of the message signal, M, is shown below. Fill in the values P and Q. c. If the digitization threshold voltage is 0V, what is the bit error rate? d. What digitization threshold voltage would minimize the bit error rate?

3 3 of 4 Problem 3. Ben Bitdiddle studies the bipolar signaling scheme from 6.02 and decides to extend it to a 4-level signaling scheme, which he calls Ben's Aggressive Signaling Scheme, or BASS. In BASS, the transmitter can send four possible signal levels, or voltages: ( 3A, A, + A, +3A), where A is some positive value. To transmit bits, the sender's mapper maps consecutive pairs of bits to a fixed voltage level that is held for some fixed interval of time, creating a symbol. For example, we might map bits 00 to 3A, 01 to A, 10 to +A, and 11 to +3A. Each distinct pair of bits corresponds to a unique symbol. Call these symbols s_minus3, s_minus1, s_plus1, and s_plus3. Each symbol has the same prior probability of being transmitted. The symbols are transmitted over a channel that has no distortion but does have additive noise, and are sampled at the receiver in the usual way. Assume the samples at the receiver are perturbed from their ideal noise-free values by a zero-mean additive white Gaussian noise (AWGN) process with noise intensity N 0 = 2σ 2, where σ 2 is the variance of the Gaussian noise on each sample. In the time slot associated with each symbol, the BASS receiver digitizes a selected voltage sample, r, and returns an estimate, s, of the transmitted symbol in that slot, using the following intuitive digitizing rule (written in Python syntax): def digitize(r): if r < -2A: s = sminus3 elif r < 0: s = sminus1 elif r < 2A: s = splus1 else: s = splus3 return s 1. The power of a symbol transmission is defined as the square of the voltage level at which the symbol is transmitted. What is the average power level, P, of a symbol transmission in BASS (i.e., the average power dissipated at the transmitter)? Ben wants to calculate the symbol error rate for BASS, i.e., the probability that the symbol chosen by the receiver was different from the symbol transmitted. Note: we are not interested in the bit error rate here. Help Ben calculate the symbol error rate by answering the following questions below from 2 through 8 2. Suppose the sender transmits symbol s_plus3. What is the conditional symbol error rate given this information; i.e., what is P(symbolError s_plus3 sent)? Express your answer in terms of A, N 0, and 2 x 2 the erfc function, defined as erfc z = e Քx. π z

4 4 of 4 3. Now suppose the sender transmits symbol s_plus1. What is the conditional symbol error rate given this information, in terms of A, N 0, and the erfc function? The conditional symbol error rates for the other two symbols don't need to be calculated separately. 4. The symbol error rate when the sender transmits symbol s_minus3 is the same as the symbol error rate of which of these symbols? 1. s_minus1. 2. s_plus1. 3. s_plus3. 5. The symbol error rate when the sender transmits symbol s_minus1 is the same as the symbol error rate of which of these symbols? 1. s_minus3. 2. s_plus1. 3. s_plus3. 6. Now suppose the sender transmits symbol s_minus1. What is the conditional symbol error rate given this information, in terms of A, N 0, and the erfc function? Do not recalculate; use symmetry and your previous answer(s). 7. Combining your answers to the questions above, what is the symbol error rate in terms of A, N 0, and the erfc function? Recall that all symbols are equally likely to be transmitted. 8. Express the answer to the above question in terms of the signal-to-noise ratio. The average signal power is in the answer to one of the questions above.

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